Neuromorphic computers based on analog neural networks aim to substantially lower computing power by reducing the need to shuttle information between memory and logic units. Artificial synapses containing analog, non-volatile conductance states enable direct computation using memory elements; however, most non-volatile analog memories require high voltages and current densities, and have nonlinear and unpredictable weight updates. Here, we develop a redox transistor based on electrochemical ion insertion into Li_xTiO₂ that not only enables linear weight updates at low current densities, but also has no baseline open-circuit voltage. The absence of an open-circuit voltage offset enables its pairing with a diffusive memristor acting as selector to create a Si-free synaptic memory cell with write voltages as low as 170 mV. We further show that the Li_xTiO₂ redox transistor can achieve an extremely sharp transistor subthreshold slope of just 40 mV/decade (at 80° C) when operating in an electrochemically driven phase transformation regime.